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Converting comment to an answer: Let $M = [0,1]$ with its usual metric (which is a length space), $X= \mathbb{R}$ and let $f : [0,1] \to \mathbb{R}$ be the Cantor function. …

Regarding a question about proofs-of-work and following up on this answer and the comments therein, I believe we can, at least in theory, come close to having the hashing resources used in cryptocurrencies … If $G_1$ and $G_2$ can be easily converted to one another, however, then the space may be smaller, and it may become harder to hash on to a long string of $0$'s. …

My question: If $M,N$ are pure pointwise pure (see Geordie Williamson's comment below), then is the induced structure on $Hom(rat(M), rat(N))$ pure? … If the answer is no, under what additional conditions (other than requiring $X$ to be smooth and complete plus $M,N$ being the `constant' sheaf) can the answer be converted to yes? …

Update #3: To appreciate why it may take awhile for a concluding summary answer to appear, please see Luca Trevisan's comment (below) that begins "By the way, your question, and my answer, do not affect … Thank you all for your patience, and thanks go to Luca Trevisan, especially, for his comments. …

Comment by Cherng-tiao Perng converted to an answer: It appears that Theorem A of this paper solves your problem. …

Synopsis and concrete practices Everyone is thanked for their comments, and in view of the diversity of views expressed, I have converted this question to a community wiki. … Therefore, instantly "accepted" will be any MathOverflow answer that provides a standardized "\hyphenation{...}" file (including mathematician's names) from any respected mathematical journal or publishing …

Converting Ofer Zeitouni's comment to an answer: Billingsley, Patrick. Convergence of Probability Measures. Ethier, Stewart and Kurtz, Thomas. Markov Processes: Characterization and Convergence. …

I am converting my comment into an answer at the request of the proposer. Given $x\in L$, let $S_x=\{y\in L\, :\, y\not\geq x\}$. …

I can show $\det A_n=\det B_n=2^{\binom{n+1}2}$; it is not a problem (see, for instance, my answer here). Question. (1) Is there a transformation converting $A_n$ into $B_n$ (or vice-versa)? … Then, $A_n$ and $B_n$ share the same Smith normal form (showing the diagonal vector) given by $$[2^{\max(4k-2n+s(n-k)-s(k),0)}:\, 1\leq k\leq n].$$ This claim is based on data from Noam Elkies' comments …

Can every $0-1$ be converted to a matrix of one component by permutations of rows and columns? What classes of matrices cannot have one component? … Christian's answer based on fedja's comment solves the problem. Infact the approach can be recursively used to get multiple component matrices for any constant number of components $k$. …

(Converting Mark Wildon's comment and mine into an answer.) Your strong version of Iwasawa's Criterion is Theorem 1.2 of the following book: Donald E. …

(In order that this question appears as answered I'm converting Richard's comment into an answer.) The (somewhat nonstandard) convention in the text, stated just before the corollary, is that $[n]! …

(Comment converted to answer per request:) The “surprising result” about simply connected homogeneous spaces in your (currently) last paragraph is Montgomery's Theorem (1950): generally (in your notation …

Let me convert my comment into an answer with references. …

Note that this is an inductive procedure for constructing PA proofs of individual instances of Theorem 1′ and cannot be converted to a PA proof of Theorem 1′ itself; however, it illustrates that each instance … to IamMeeoh's mathoverflow question—my comment])? …